Problem 18 - Calculus

Question

Sketch the curves described in (a)-(c): (a) Slope is positive and increasing at first but then is positive and decreasing. (b) The first derivative of the function whose graph is in part (a). (c) The second derivative of the function whose graph is in part (a).

Step-by-Step Solution

Answer

(a) Sketch a cubic curve; (b) Its first derivative is a downward parabola; (c) The second derivative is a straight line.

Step 1: Understanding Part (a) - The Curve Description

For part (a), we need a curve where the slope is positive and increasing initially but eventually becomes positive and decreasing. This typically describes a curve that first becomes steeper and then less steep. An example curve could be the increasing part of a cubic function.

Step 2: Sketching the Curve for Part (a)

On a graph, draw a curve that starts from the origin and initially appears similar to a steep hill, getting progressively steeper. As it reaches the peak, it maintains an upward movement but becomes gradually less steep, suggesting it is still rising but at a slower rate. A perfect example is the function such as \( y = x^3 - x \) for a specific range of \( x \).

Step 3: Understanding Part (b) - First Derivative

The first derivative of a function represents its slope or rate of change. For the graph in (a), the first derivative must be positive and first increase ('steeper') and then decrease ('less steep'). A quadratic function can represent this as its graph is a parabola that opens downward.

Step 4: Sketching the First Derivative - Part (b)

Sketch a downward-facing parabola. This represents the first derivative of the original cubic-like function curve graph. For example, if the curve in (a) is \( y = x^3 - x \), its derivative is \( y' = 3x^2 - 1 \), which is a downward-opening parabola.

Step 5: Understanding Part (c) - Second Derivative

The second derivative represents the concavity or the rate of curvature change in the original function. Since we have a parabolic shape for our first derivative, the second derivative would be a linear equation.

Step 6: Sketching the Second Derivative - Part (c)

Draw a straight line going downward on a graph. For the cubic function example \( y = x^3 - x \), the second derivative would be \( y'' = 6x \), which is a straight line through the origin with a positive slope.

Key Concepts

First Derivative

When we talk about the first derivative of a function, we're focused on understanding how the slope or steepness of the curve changes over time. Imagine you're hiking up a hill. The first derivative tells us how steep the hill is at any given point along your path.
You can think of the first derivative as the function that gives us the slope of the original curve. For part (a) of the exercise, we want to sketch a function that is steep at first and then becomes less steep. This kind of curve is typically represented by a cubic function like \( y = x^3 - x \). When we derive this function, we get \( y' = 3x^2 - 1 \).

• The quadratic function \( 3x^2 - 1 \) shows the slope of the original curve, which starts high (positive and increasing) and then drops (positive but decreasing).
• Graphically, this is shown as a downward-facing parabola, meaning our slope starts big and gradually lessens.

Understanding the first derivative helps us sketch the behavior of the original function by showing where it rises sharply and where it starts leveling off.

Second Derivative

The second derivative gives us insights into a curve's concavity, or how it "bends." It helps indicate whether our curve is smiling or frowning at different parts of our graphical journey. If you've ever had to push a ball up a hill and noticed how, at first, it resists by rolling back, then at the top, it starts to head down the other way, you've witnessed the effects of concavity.
In exercise part (c), we need the second derivative to examine the curve's rate of change in curvature. If we work with the function \( y = x^3 - x \), the second derivative is \( y'' = 6x \).

• This equation describes a straight line passing through the origin, giving us critical information on the acceleration of the slope at different points.
• A positive slope in this straight line implies our steepness is increasing as we move forward.

By understanding the second derivative, we can predict whether our original curve is opening upwards or downwards as we sketch it.

Cubic Function

Cubic functions are a fascinating type of polynomial equation where the highest degree is three. They take the form \( ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a eq 0 \).
These functions can create curves that change direction, which makes them perfect for modeling real-world scenarios like roller coaster tracks or the speed of a car over time.

• With cubic functions, you may see a rise, fall, and then rise again as you move along the curve, which adds a level of complexity over simpler quadratic functions.
• The behavior of the cubic function largely depends on the coefficients, particularly \( a \), as it dictates how 'wide' or 'narrow' the cubic curve appears.

The example \( y = x^3 - x \) is an excellent demonstration of a cubic function. It captures the essence of what makes these functions so intriguing—let's picture it by considering its real-valued roots and inflection points that occur based on its derivatives.
By grasping the cubic function, you can uncover many layers of a graph's story, predicting and understanding changes with each step along the way.